Calculating Present Value for Investment

How much money should be deposited today in an account that earns 8 % compounded semiannually so that it will accumulate to $ 15 comma 000 in three​ years? LOADING... Click the icon to view some finance formulas. Question content area bottom Part 1 The amount of money that should be deposited is ​$    enter your response here. ​(Round up to the nearest​ cent.)

To find out how much money should be deposited today (the present value, ( PV )) in an account that earns 8% interest compounded semiannually, we can use the formula for compound interest:

[ FV = PV \times (1 + r/n)^{nt} ]

Where:

• ( FV ) is the future value (the amount we want to accumulate, which is $15,000),
• ( PV ) is the present value (the amount we need to find),
• ( r ) is the annual interest rate (8% or 0.08),
• ( n ) is the number of times interest is compounded per year (2 for semiannual),
• ( t ) is the number of years the money is invested (3 years).

We need to rearrange the formula to solve for ( PV ):

[ PV = \frac{FV}{(1 + r/n)^{nt}} ]

Now, substituting the values into the formula:

• ( FV = 15000 )
• ( r = 0.08 )
• ( n = 2 )
• ( t = 3 )

Calculating ( PV ):

1. Calculate ( r/n ): [ r/n = 0.08/2 = 0.04 ]

2. Calculate ( nt ): [ nt = 2 \times 3 = 6 ]

3. Calculate ( (1 + r/n)^{nt} ): [ (1 + 0.04)^{6} = (1.04)^{6} ]

   Now calculate ( (1.04)^{6} ): [ (1.04)^{6} \approx 1.265319 ]

4. Now substitute back into the formula for ( PV ): [ PV = \frac{15000}{1.265319} \approx 11861.45 ]

Thus, the amount of money that should be deposited today is approximately $11,861.45.

Final Answer: The amount of money that should be deposited is $11,861.45 (rounded to the nearest cent).